Overview
This work investigates determinantal representations for higher secant varieties of smooth projective curves. Specifically, it establishes the existence of symmetric admissible determinantal representations for these varieties when embedded with sufficient ampleness. These representations are associated with symmetric Ulrich sheaves of rank one. The research also addresses conditions for positive definiteness of representing matrices in the context of secant varieties of real curves. Furthermore, it provides explicit representations for secant varieties of rational normal curves.
Approach
The research establishes that higher secant varieties of smooth projective curves, under conditions of sufficiently ample embedding, admit symmetric admissible determinantal representations. These representations incorporate symmetric Ulrich sheaves of rank one. For secant varieties of real curves, the study identifies specific conditions that lead to positive definite representing matrices. This enables, under specific conditions, the representation of the convex hull of the curve as a spectrahedron, particularly when it functions as a hyperbolicity cone. The investigation into secant varieties of rational normal curves utilizes Littlewood-Richardson coefficients to derive explicit representations. A fundamental tool employed in this methodology is the application of higher Szegő kernels and higher Scorza correspondences, which are associated with a non-effective theta characteristic on the curve.
Findings
- Higher secant varieties of smooth projective curves have symmetric admissible determinantal representations with symmetric Ulrich sheaves of rank one, given sufficient ampleness of the embedding.
- For secant varieties of real curves, conditions are outlined for the representing matrices to be positive definite. This allows for the representation of the convex hull of the curve as a spectrahedron when it is a hyperbolicity cone, under certain conditions.
- Very explicit representations for secant varieties of rational normal curves are derived using Littlewood-Richardson coefficients.
- The methodology leverages higher Szegő kernels and higher Scorza correspondences, linked to a non-effective theta characteristic on the curve.